Optimisation of Optimisation of Irradiation DirectionsIrradiation Directions

in IMRT Planningin IMRT Planning

Rick JohnstonRick JohnstonMatthias EhrgottMatthias EhrgottDepartment of Engineering ScienceDepartment of Engineering ScienceUniversity of AucklandUniversity of Auckland M. Ehrgott, R. Johnston Optimisation of M. Ehrgott, R. Johnston Optimisation of

Irradiation Directions in IMRT Planning, OR Irradiation Directions in IMRT Planning, OR Spectrum 25(2):251-264, 2003Spectrum 25(2):251-264, 2003

What is Radiotherapy?What is Radiotherapy?

Intensity modulation - Intensity modulation - improves treatment improves treatment quality quality

Inverse planning problem - Inverse planning problem - conflicting conflicting

objectives toobjectives to irradiate tumour without damage to irradiate tumour without damage to healthy organshealthy organs

IMRTIMRT

Model FormulationModel Formulation

Discretisation of Body and BeamDiscretisation of Body and Beam

gantry

VoxelsVoxels

BixelsBixels

Angle DiscretisationAngle Discretisation

Linearises the Linearises the problemproblem

A number of LPs A number of LPs to be solved to be solved

Replicates Replicates physical setupphysical setup

MOMIP ModelMOMIP Model

DataData LL11 == lower bound in tumourlower bound in tumour UUkk == upper bound in organ upper bound in organ kk RR = number of directions to be used = number of directions to be used

Variables and functionsVariables and functions Intensity vector Intensity vector x = (xx = (x1111,...,x,...,xHNHN)) Direction choice vector Direction choice vector y = (yy = (y11,...,y,...,yHH)) Deviation vector Deviation vector T = (TT = (T11,...,T,...,TKK)) Dose distribution vectors Dose distribution vectors DDkk = (D = (Dk1k1,...,D,...,DkMkkMk))

min (Tmin (T11,...,T,...,TKK))

DD11 = P = P11x x (L (L1 1 - T- T11))11

DDk k = P= Pkkx x (U (Uk k + T+ Tkk))11, k=2,...,K, k=2,...,K xxhihi My Myhh, h=1,…,H, i=1,…,N, h=1,…,H, i=1,…,N

yy11+ ...+y+ ...+yHH R R

yyh h {0,1} h=1,...,H {0,1} h=1,...,H

T, x T, x 00

To study effect of direction optimisation consider To study effect of direction optimisation consider weighted sum min weighted sum min 11TT11+ + 22TT22 + ... + + ... + KKTTKK

Extension of multicriteria model by Hamacher/KüferExtension of multicriteria model by Hamacher/Küfer

Solution MethodsSolution Methods

Two-phase MethodsTwo-phase Methods

3. Set Covering3. Set Covering

4. LP Relaxation4. LP Relaxation

Integrated MethodsIntegrated Methods

1. Mixed Integer 1. Mixed Integer FormulationFormulation

2. Local Search 2. Local Search Heuristics Heuristics

Integrated MethodsIntegrated Methods

CPLEX 7.0CPLEX 7.0 If If RR increases problem becomes increases problem becomes

easier, objective value improveseasier, objective value improves For small For small RR and small angle and small angle

discretisation often no feasible discretisation often no feasible solution foundsolution found

MIP SOLVER1

Optimal solution of MIP problemOptimal solution of MIP problem

Isodose curve pictures obtained with Isodose curve pictures obtained with prototype software developed at prototype software developed at ITWM, KaiserslauternITWM, Kaiserslautern

Integrated MethodsIntegrated Methods

Alter each gantry position in turn to find Alter each gantry position in turn to find better anglesbetter angles

Steepest descent with randomised starting Steepest descent with randomised starting anglesangles

Solve LP for each selection of anglesSolve LP for each selection of angles

LOCAL SEARCH2

Local Search MovieLocal Search Movie

Two-phase MethodsTwo-phase Methods

IntuitiveIntuitive Considers all anglesConsiders all angles Relatively quickRelatively quick

Fully irradiate every voxel in the tumour

Avoid damage to healthy organs

Benefits:Benefits:

SET COVERING3

min min CC11yy11+...++...+CCSSyySS

AyAy 11

yy {0,1}{0,1}

aaijij=1 if and only if =1 if and only if beam beam jj hits voxel hits voxel i i

Weighted angle methodWeighted angle method

CCjj is sum of is sum of kk/U/Ukk over all organs over all organs at risk and voxels in beam at risk and voxels in beam jj

Dose deposition methodDose deposition method

CCjj is sum of is sum of kkPPkk(i,j)/U(i,j)/Ukk over all over all voxels and all organs at riskvoxels and all organs at risk

Cost coefficientsCost coefficients

Com parison of Cost Coefficients

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340

An gle (degrees)

Cost(WeightedAngle)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Cost(DoseDeposition)

W eighted Angle M ethod

Dose Deposition M ethod

Set Covering SolutionSet Covering Solution MIP SolutionMIP Solution

4 Two-Phase MethodsTwo-Phase MethodsLP RELAXATIONLP RELAXATION

Optimal solution ofOptimal solution ofLP relaxationLP relaxation10-40 beams used10-40 beams used

ResultsResults

All methods were successful in All methods were successful in generating good treatment plans in a generating good treatment plans in a reasonable timeframe (10 min)reasonable timeframe (10 min)

Optimal beams were Optimal beams were often counterintuitiveoften counterintuitive

Angle optimisation is Angle optimisation is important if few beams important if few beams to be usedto be used

Solution with Solution with equidistant beamsequidistant beams

Solution with Solution with optimised beamsoptimised beams

ComparisonComparisonO

bje

ctiv

e

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Problem 1 3 heads Problem 1 4 heads Problem 2 3 heads Problem 2 4 heads Problem 3 3 heads

Set CoveringLP relaxationLocal SearchMixed Integer

Objective vs. TimeObjective vs. TimeO

bje

ctiv

e

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 2000 4000 6000 8000 10000 12000

Time (s)

Local search improvement

Set Covering

Local Search

LP relaxation

Mixed Integer